具有各向同性相的双组分复合材料场递推法的严格逼近
A rigorous approach to the field recursion method for two-component composites with isotropic phases
摘要：
在这一章的书，题为“扩展复合材料理论的其他领域的科学”[编辑格雷姆W。Milton，2016]我们给出了复合材料抽象理论中场方程递归方法对具有各向同性相的双组分复合材料的严格推导。这种方法是非常感兴趣的，因为它已被证明是一个强大的工具，在开发尖锐的边界的有效张量的复合材料。其原因是有效张量$\bf L_*$可以在复合抽象理论的一般框架中解释为某个正交$Z(2)$子空间集合上的$Z$ -算子。递归的基本情况开始于希尔伯特空间$\cal H$上的正交$Z(2)$子空间集合、$Z$问题和相关的$Y$问题。我们给出了$Z$ -问题和相关的$Y$ -问题的可解性的一些新条件.给出了相应的$Z$ -算子和$Y$ -算子的显式表示，并研究了它们的解析性质.一个迭代方法，然后开发从一个层次的子空间集合及其相关的运营商，导致连续的分数表示的初始有效张量$\bf L_*$。
英文摘要：
In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we give a rigorous derivation of the field equation recursion method in the abstract theory of composites to two-component composites with isotropic phases. This method is of great interest since it has proven to be a powerful tool in developing sharp bounds for the effective tensor of a composite material. The reason is that the effective tensor $\bf L_*$ can be interpreted in the general framework of the abstract theory of composites as the $Z$-operator on a certain orthogonal $Z(2)$ subspace collection. The base case of the recursion starts with an orthogonal $Z(2)$ subspace collection on a Hilbert space $\cal H$, the $Z$-problem, and the associated $Y$-problem. We provide some new conditions for the solvability of both the $Z$-problem and the associated $Y$-problem. We also give explicit representations of the associated $Z$-operator and $Y$-operator and study their analytical properties. An iteration method is then developed from a hierarchy of subspace collections and their associated operators which leads to a continued fraction representation of the initial effective tensor $\bf L_*$.
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